Two more key references in this answer: the Itzykson & Zuber discussion for the σ-model alluded to is peerless, if one really wishes to get to the bottom of it; summarized here. The chiral model calculation is trivial and I am illustrating it below in a thus retargetted question. (The subtler point in your question, however, is connecting quark bilinear currents to pseudoscalar hadrons, a crepuscular art, and the sheer genius brainchild of Murray Gell-Mann, long before the discovery of confinement! His Escoffier pheasant baked between veal slices picture, glossed over here. This is a hard question, split off from the body of yours, requiring subtle verbiage: the core of M G-M's historic tie-breaker to Renner, above.)
Here, then, is the trivial answer of the retargeted question merely illustrating the operator PCAC statement with Gürsey's 1960 spectacular chiral model for $SU(2)_L\times SU(2)_R$ for you to get the basic idea; whose 3-flavor extension is alarmingly called the "Goldstone matrix" at the end of the talk you linked. I do this since the Pauli matrices for SU(2) are better understood intuitively, and I am terminally cavalier with normalizations, using ~ signs instead of =, since you should appreciate the structure of the representations and how they fit, and not the details utilized in monkey-see-monkey-do calculations.
Again, I am eschewing the $64 question of hadronizing the quark operators for which subtler current algebra "hypothesis" arguments were formulated by Gell-Mann--Oakes--Renner, as alluded to in the comments. So everything I'll cover will be solid operator statements in the context of that model, strict equalities, without any hypotheses and pheasants -- the fast talking involved bridging to QCD. It is important below, to illustrate all current statements with how the currents also present in the linear σ model, covered brilliantly in the textbooks cited; and also the quark bilinears of QCD. Both these are alternate realizations of the same group structure illustrated here.
Start with CVC and CAC, for the ideal, merely SSBroken chiral model of Goldstone mesons, without explicit breaking (which does not affect the currents!), yet,
$$
{\cal L} \sim \frac{f_\pi^2}{4} \operatorname{Tr} ~ \partial^\mu U^\dagger \partial_\mu U , \\
U= e^{i \vec \pi\cdot \vec \tau /f_\pi} = 1\!\! 1 \cos (|\pi|/f_\pi) + i \frac{\vec \pi \cdot \vec \tau}{|\pi|}\sin (|\pi|/f_\pi),\\
|\pi|\equiv \sqrt{\vec \pi \cdot \vec \pi} .
$$
It is then evident all four currents are conserved,
$$
{\vec L}_\mu = if_\pi^2 \operatorname {Tr}~ \vec\tau ~ U\partial_\mu U^\dagger/2 \sim f_\pi \partial_\mu \vec \pi +..., \\
\vec R_\mu = if_\pi^2 \operatorname {Tr}~ \vec\tau ~ U^\dagger \partial_\mu U /2 \sim -f_\pi \partial_\mu \vec \pi +..., \\
\vec V_\mu =\vec L_\mu+\vec R_\mu= if_\pi^2 \operatorname {Tr}~ \vec\tau ~ (U\partial_\mu U^\dagger +U^\dagger \partial_\mu U )/2 \sim -2f_\pi \vec \pi \times \partial_\mu \vec \pi +... \\
\vec A_\mu = \vec R_\mu-\vec L_\mu= if_\pi^2 \operatorname {Tr}~ \vec\tau ~ (U\partial_\mu U^\dagger -U^\dagger \partial_\mu U )/2 \sim -2f_\pi \partial_\mu \vec \pi +... ,
$$
that is, L is the right-invariant left SU(2) current, R the left-invariant right SU(2) current, V the vector/isospin current, and A the axial current your question asks about. The key point here is that only V escapes SSB and acts linearly, in the Wigner-Weyl mode: its charge isorotates Goldstone mesons.
The remaining three, L,R,A are in the Nambu-Goldstone nonlinear mode: their charges shift goldstons into and out of the (therefore) degenerate vacuum.
This is what SSB means.
- It is a worthwhile extraneous exercise to work out the above currents in both the linear σ model (the original ambit of these observations) and the free quark QCD lagrangian, and compare, while also confirming the standard current algebra for them. Crudely, $V_\mu\sim \vec \pi \times \partial_\mu \vec \pi$, $\sim \bar q \gamma_\mu \vec \tau q$; and $A_\mu\sim \sigma \partial_\mu \vec \pi+...$, $\sim \bar q \gamma_\mu \gamma_5 \vec \tau q$.
For the above completely symmetric lagrangian, the currents are strictly conserved, and we have CVC and CAC.
However, a major triumph in the 60s and the σ model was the tapered mooting of CAC into PCAC by adding some explicit breaking (the σ term) in the lagrangian. In the above chiral model at hand, this amounts to something like
$$
-2\frac{m_\pi^2 f_\pi^2}{\operatorname {Tr}M} \operatorname {Tr}~(MU+U^\dagger M),
$$
where M is a matrix despoiling L and R invariance explicitly (and identified today with a diagonal u,d quark mass matrix--don't worry about it; you might as well take it to be the identity) and the dimensional parameters in front of it are chosen with the benefit of hindsight. Since it has no derivatives, it contributes nothing to the above currents.
Expanding in pions, this amounts to a constant and, importantly, a pion mass term, $m^2_\pi \vec \pi \cdot \vec \pi$.
This mass term converts the massive pions into "pseudogoldstone bosons", and, by dint of the modified equations of motion, it leads to a non vanishing divergence for the axial current,
$$
\partial^\mu \vec A _\mu \sim f_\pi m_\pi^2 \vec \pi +..., \tag{#}
$$
the operator PCAC fact, not hypothesis, here (only).
Taking matrix elements between the vacuum and the pion states it destroys, trivially yields the matrix relations $\langle 0|\partial \cdot A^a(0)|\pi^b\rangle \sim f_\pi m_\pi^2 \delta^{ab}$.
The imaginative leap of Gell-Mann (with a nudge by Feynman$^\dagger$) was to correctly (as most often!) abstract the essential logical structure of the picture and hypothesize that the matrix element relations also held for quark (now QCD) chiral currents, in general, as well.
Spot on. 60 years later, QCD flows in the logical channels then dug.
$\dagger$ RPF is credited with discovering (#) in the historic paper which introduced it. For the history maven, cf. here. The deeper, more general "hypothesis" version involving nucleons appears in R Norton & W Watson, Phys Rev 110 (1958) 996.
